Rigidity of spacelike translating solitons in pseudo-Euclidean space
aa r X i v : . [ m a t h . DG ] A p r Rigidity of spacelike translating solitons inpseudo-Euclidean space ∗ Ruiwei Xu Tao Liu
Abstract:
In this paper, we investigate the parametric version and non-parametricversion of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space R m + nn . Firstly, we classify m -dimensional complete spacelike translating solitons in R m + nn by affine technique and classical gradient estimates, and prove the only com-plete spacelike translating solitons in R m + nn are the spacelike m -planes. This resultprovides another proof of a nonexistence theorem for complete spacelike translatingsolitons in [8], and a simple proof of rigidity theorem in [33]. Secondly, we general-ize the rigidity theorem of entire spacelike Lagrangian translating solitons in [34] tospacelike translating solitons with general codimensions. As a directly application oftheorem, we obtain two interesting corollaries in terms of Gauss image. Keywords: translating soliton; rigidity theorem; pseudo-Euclidean space.
The mean curvature flow (MCF) in Euclidean space (pseudo-Euclidean space resp.) isa one-parameter family of immersions X t = X ( · , t ) : M m → R m + n ( R m + nn resp. )with the corresponding image M t = X t ( M ) such that ddt X ( x, t ) = H ( x, t ) , x ∈ M,X ( x,
0) = X ( x ) , (1.1)is satisfied, here H ( x, t ) is the mean curvature vector of M t at X ( x, t ) in R m + n ( R m + nn resp.). The MCF in higher codimension has been studied extensively in the last few ∗ Research partially supported by NSFC. M m is said to be a translatingsoliton in R m + n (spacelike translating soliton in R m + nn resp.) if the mean curvaturevector H satisfies H = T ⊥ , (1.2)where T ∈ R m + n is a non-zero constant vector, which is called a translating vector,and T ⊥ denotes the orthogonal projection of T onto the normal bundle of M . M m issaid to be a self-shrinker in R m + n (spacelike self-shrinker in R m + nn resp.) if it satisfies aquasi-linear elliptic system H = − X ⊥ , (1.3)where X ⊥ is the normal part of X , which is an important class of solutions to (1.1).There is a plenty of works on the classification and uniqueness problem for trans-lating soliton and self-shrinker in Euclidean space (cf. [2, 4, 9, 11, 14, 18, 21, 23, 25, 27]).On the other hand, there are many works on the rigidity problem for complete spacelikesubmanifolds in pseudo-Euclidean space. Calabi [3] proposed the Bernstein problemfor spacelike extremal hypersurfaces in Minkowski space R m +11 and proved such hyper-surfaces have to be hyperplanes when m ≤
4. In [10] Cheng-Yau solved the problemfor all dimensions. Later, Jost-Xin [20] generalized the results to higher codimensions.
Theorem 1 [20]:
Let M m be a spacelike extremal submanifold in R m + nn . If M m is closed with respect to the Euclidean topology, then M has to be a linear subspace. Translating soliton and self-shrinker can be regarded as generalizations of extremalsubmanifold. Thus it is natural to study the corresponding rigidity problem for space-like translating soliton and self-shrinker. Here we only investigate the rigidity of space-like translating soliton in R m + nn .For complete spacelike Lagrangian translating solitons in R nn , Xu-Huang [33]proved the following rigidity theorem. Theorem 2 [33]:
Let f ( x , ..., x n ) be a strictly convex C ∞ -function defined ona convex domain Ω ⊂ R n . If the graph M ∇ f = { ( x, ∇ f ( x )) } in R nn is a completespacelike translating soliton, then f ( x ) is a quadratic polynomial and M ∇ f is an affine n -plane. From Theorem 2 above, it is easy to see that the corresponding translating vectormust be spacelike . In [8], Chen-Qiu proved a nonexistence theorem for complete space-like translating solitons in R m + nn by establishing a very powerful generalized Omori-Yaumaximum principle. They proved that there exists no complete m -dimensional spaceliketranslating soliton (with a timelike translating vector) . In fact, such nonexistence con-clusion still holds for the lightlike translating vector case. Motivating by papers [8, 33],2e classify m -dimensional complete spacelike translating solitons in R m + nn by affinetechnique and classical gradient estimates, and obtain the following Bernsterin theo-rem. Theorem 3:
Let M m be a complete spacelike translating soliton in R m + nn , then itis an affine m -plane. A more precise statement of the assertion in Theorem 3 says that there exists an m -dimensional complete spacelike translating soliton in R m + nn only if the translating vectoris spacelike . It provides another proof of the nonexistence theorem of translatingsoliton in [8], and a simple proof of the Bernstein theorem for translating soliton in [33].Here we mention that it is valid if one shall use the generalized Omori-Yau maximumprinciple in [8] to prove Theorem 3 above.For non-parametric Lagrangian solitons of mean curvature flow, there are severalinteresting rigidity theorems. As to entire self-shrinker for mean curvature flow in R nn with the indefinite metric P ni =1 dx i dy i , Huang-Wang [16] and Chau-Chen-Yuan [5] useddifferent methods to prove the rigidity of entire self-shrinker under a decay condition onthe induced metric ( D f ). Later, using an integral method, Ding and Xin [13] removedthe additional decay condition and proved any entire smooth convex self-shrinkingsolution for mean curvature flow in R nn is a quadratic polynomial. In [17], Huangand Xu investigated the rigidity problem of entire spacelike translating soliton graph( x, ∇ f ) in R nn with some symmetry conditions. Motivating by [5,13,16,17], Xu-Zhu [34]proved a rigidity theorem of entire convex translating solutions for mean curvature flowin R nn under a decay condition on the induced metric ( D f ) and provided a class ofnontrivial entire spaclike Lagrangian translating solitons. Theorem 4 [34]:
Let f ( x ) be an entire smooth strictly convex function on R n ( n ≥
2) and its graph M ∇ f = { ( x, ∇ f ( x )) } be a translating soliton in R nn . If thereexists a number ǫ > D f ) satisfies( D f ) > ǫ | x | I, as | x | → ∞ , (1.4)then f ( x ) must be a quadratic polynomial and M ∇ f is an affine n -plane.Here we shall use the idea in [34] to generalize Theorem 4 to spacelike graphictranslating solitons in pseudo-Euclidean space R m + nn . Theorem 5:
Let u α (1 ≤ α ≤ n ) be smooth functions defined everywhere in R m and their graph M = ( x, u ( x ) , u ( x ) , · · · , u n ( x )) be a spacelike translator in R m + nn . Ifthere exists a number ǫ > g ij ) satisfies( g ij ) > ǫ | x | I, as | x | → ∞ , (1.5)3hen u ( x ) , · · · , u n ( x ) are linear functions on R m , and M is an affine m -plane in R m + nn . Remark : It is necessary that there is a restriction on the induced metric ( g ij ) forthe rigidity of translating solitons. If not, there exist nontrivial entire smooth spaceliketranslating solitons, which are not planes. For example, submanifold( x, y, ln(1 + exp { x } ) − x, µy ) , | µ | < , ( x, y ) ∈ R (1.6)is an entire spacelike translator with the translating vector (0 , , , µ ) in R , whichgraphic functions satisfy the PDE (2.11).By studying the distribution of the Gauss map, they obtained Bernstein theoremsof translating solitons in Euclidean space (see [2], [21] and [32]). Notice that the Gaussimage of spacelike graphic submanifold M m in R m + nn is bounded if and only if theinduced metric g = det( g ij ) is bounded (see [31]). Therefore, it is easy to see thatexample (1.6) above and example (1.7) in [34] have boundless Gauss images. As adirectly application of theorem 5, we have Corollary 1:
Let M m be an entire spacelike graphic translator in R m + nn as definedin Theorem 5. If the Gauss image of M is bounded, then M is an affine m -plane.By relaxing the bound of the Gauss image to controlled growth, we also get amore general corollary from theorem 5 as Prof. Dong generalized a rigidity theoremfor spacelike graph with parallel mean curvature in [15]. Corollary 2:
Let M m be an entire spacelike graphic translator in R m + nn as definedin Theorem 5. If there exists a number ǫ > g ij ) satisfiesdet( g ij ) > ǫ | x | , as | x | → ∞ , (1.7)then M is an affine m -plane. The pseudo-Euclidean space R m + nn is the linear space R m + n endowed with the metric ds = m X i =1 ( dx i ) − m + n X α = m +1 ( dx α ) . (2.1)Let X : M m → R m + nn be a spacelike m -submanifold in R m + nn with the second funda-mental form B defined by B Y W := ( ∇ Y W ) ⊥ (2.2)for Y, W ∈ Γ( T M ), where ∇ denotes the connection on R m + nn . Let ( · ) ⊤ and ( · ) ⊥ denote the orthogonal projections into the tangent bundle T M and the normal bundle4 M , respectively. Let ∇ and ∇ ⊥ be connections on the tangent bundle and thenormal bundle of M , respectively. Choose a local Lorentzian frame field { e i , e α } ( i =1 , · · · , m ; α = m + 1 , · · · , m + n ) such that { e i } are tangent vectors to M . The meancurvature vector of M in R m + nn is defined by H = X α H α e α = X i B ii = − X i,α h αii e α , (2.3)where h αii = h B ii , e α i . Here, h· , ·i is the canonical inner product in R m + nn . We have theGauss equation R ijkl = − X α ( h αik h αjl − h αil h αjk ) , (2.4)and the Ricci curvature R ij = − X α,k ( h αkk h αij − h αki h αkj ) . (2.5)For a spacelike translating soliton M m , by definition we can decompose the trans-lating vector T into a tangential part V and a normal part H on M m , namely T = V + H . Define k H k = −h H, H i = −| H | , where k H k is absolute value of the norm square of the mean curvature. Similarlydefine k B k = −h B, B i = −| B | . From (2.3) and the inequality of Schwartz, we have k B k ≥ m k H k . (2.6)Note that when the spacelike manifold M m is a Lagrangian gradient graph ( x, ∇ f ) in R nn with the indefinite metric P ni =1 dx i dy i , the functions k H k and k B k are the normof Tchebychev vector field Φ and the Pick invariant J in relative geometry respectively(see [33]), up to a constant. Therefore we can use some affine technique to estimatefunctions k H k and k B k .Let z = h X, X i be the pseudo-distance function on M . Then we have (see also [30]) z ,i = e i ( z ) = 2 h X, e i i , (2.7) z ,ij = Hess ( z )( e i , e j ) = 2( δ ij − h X, h αij e α i ) , (2.8)∆ z = X z ,ii = 2 m + 2 h X, H i . (2.9)In the following we set up the basic notations and formula for an m -dimensionspacelike graphic submanifold in R m + nn . Let M := { ( x , · · · , x m , u , · · · , u n ); x i ∈ R , u α ( x ) = u α ( x , · · · , x m ) } , i = 1 , · · · , m and α = 1 , · · · , n . Denote x = ( x , · · · , x m ) ∈ R m ; u = ( u , · · · , u n ) ∈ R n . Let E A ( A = 1 , · · · , m + n ) be the canonical Lorentzian basis of R m + n . Namely, everycomponent of the vector E A is 0, except that the A -th component is 1. Then e i = E i + X α u αi E m + α , i ∈ { , · · · , m } give a tangent frame on M . Here, u αi = ∂u α ∂x i . In pseudo-Euclidean space with index n ,the induced metric on spacelike submanifold M is g ij = h e i , e j i = δ ij − X α u αi u αj . Then there are n linear independent unit normal vectors, e α = P i u αi E i + E m + α (1 − | Du α | ) , α ∈ { , · · · , n } , where Du α = ( u α , · · · , u αm ). Thus the Levi-Civita connection with respect to theinduced metric has the Chistoffel symbolsΓ kij = 12 g kl ( ∂g il ∂x j + ∂g jl ∂x i − ∂g ij ∂x l ) = − g kl u αij u αl . (2.10)Put a nonzero constant vector T := X a i E i + X b α E m + α . By the definition of the translator (1.2), for each e α , there holds h T, e α i = − H α . By calculation, we have h H, e α i = g ij h∇ e i e j , e α i = g ij h u βij E m + β , e α i = − g ij u αij (1 − | Du α | ) , and h T, e α i = a i u αi − b α (1 − | Du α | ) . Then (1.2) is equivalent to the following elliptic system g ij u αij = − a i u αi + b α , α ∈ { , · · · , n } . (2.11)6 Calculation of ∆ k H k and ∆ k B k Proposition 3.1:
For the spacelike translating soliton M m in R m + nn , the followingestimate holds ∆ k H k ≥ m k H k − h T, ∇k H k i . Proof:
Let { e i } be a local orthonormal normal frame field at the considered pointof M . From (1.2), we derive ∇ ⊥ e j H = ( ∇ e j H ) ⊥ = ∇ e j T − X k h T, e k i e k !! ⊥ = − X k h T, e k i B jk , (3.1)and ∇ ⊥ e i ∇ ⊥ e j H = − X k h T, e k i∇ ⊥ e i B jk − X k h H, B ik i B jk . (3.2)Hence, using the Codazzi equation ∇ ⊥ e i B jk = ∇ ⊥ e k B ji , we have ∆ | H | = 2 |∇ ⊥ H | + 2 h H, ∆ ⊥ H i = 2 |∇ ⊥ H | − h H, ∇ ⊥ V H i − X i,k h H, B ik i . (3.3)It follows that ∆ k H k = 2 k∇ ⊥ H k + 2 h H, ∇ ⊥ V H i + 2 X i,k h H, B ik i ≥ h H, ∇ ⊥ V H i + 2 m k H k . (3.4)Note that 2 h H, ∇ ⊥ V H i = ∇ V h H, H i = −∇ V k H k = −h T, ∇k H k i . (3.5)Then (3.4) and (3.5) together give proposition 3.1. (cid:3) In order to prove the completeness of entire graph spacelike translator with respectto the induced metric, we need the following type estimate for ∆ k B k . Proposition 3.2:
For the spacelike translating soliton M in R m + nn , the Laplacianof the second fundamental form B satisfies∆ k B k ≥ n k B k − h T, ∇k B k i . roof: Let { e i } be a local tangent orthonormal frame field on M and { e α } alocal normal orthonormal frame field on M such that ∇ e α = 0 at the considered point.From [30], we have∆ k B k = ∆ X α,i,j ( h αij ) = 2[ X ( h αij,k ) + h αij h αkk,ij − h αij h αli h βlj h βkk + h αij h βij h αlk h βlk + ( h αij h βik − h βlj h αkl )( h αqj h βqk − h βqj h αkq )] . (3.6)Let S αβ = X i,j h αij h βij , N ( h α ) = X i,j ( h αij ) , then k B k = X α S αα . So (3.6) becomes ∆ k B k = 2[ X ( h αij,k ) − h αij H α,ij + h αij h αli h βlj H β + X α,β ( S αβ ) + N ( h α h β − h β h α )] . (3.7)Note that X α,β ( S αβ ) ≥ n ( X α S αα ) = 1 n k B k ,N ( h α h β − h β h α ) ≥ . From (3.2) and Codazzi equation, we get X α,i,j h αij H α,ij = X i,j h B ij , ∇ ⊥ e i ∇ ⊥ e j H i = − X h B ij , ∇ ⊥ V B ij i − X i,j,k h H, B ik ih B ij , B jk i = − h V, ∇k B k i + X h αij h αjk h βik H β . Then substituting these inequalities into (3.7) completes the proof of proposition 3.2. (cid:3)
To gain Bernstein theorem, we are about to prove k H k ≡ M . If k H k = 0, thenthere exists a point p ∈ M such that k H k ( p ) > . Set k H k ( p ) = λ . Denote by r ( p , p ) the geodesic distance function from p ∈ M with respect to the induced metric g . For any positive number a , let B a ( p ) := { p ∈ M | r ( p , p ) ≤ a } . We consider thefunction Ψ := ( a − r ) k H k (4.1)8efined on B a ( p ). Obviously, Ψ attains its supremum at some interior point p ∗ . Wemay assume that r is a C -function in a neighborhood of p ∗ . Choose an orthonormalframe field on M around p ∗ with respect to the metric g . Then, at p ∗ , we have ∇ Ψ = 0 , ∆Ψ ≤ . Hence − r ) ,i a − r + ( k H k ) ,i k H k = 0 , (4.2) − r a − r − |∇ r | ( a − r ) + ∆ k H k k H k − ( ∇k H k ) k H k ≤ , (4.3)where ” , ” denotes the covariant derivative with respect to the metric g .Inserting Proposition 3.1 into (4.3), we get − r a − r − |∇ r | ( a − r ) − h T, ∇k H k ik H k + 2 m k H k − ( ∇k H k ) k H k ≤ . (4.4)Combining (4.2) with (4.4), we have0 ≥ − r a − r − r ( a − r ) − h T, ∇k H k ik H k + 2 m k H k . (4.5)By (4.2) and the inequality of Schwarz, we obtain h T, ∇k H k k H k i = h V, ∇k H k k H k i≤ ǫ | V | + 4 ǫ r ( a − r ) = ǫ ( C + k H k ) + 4 ǫ r ( a − r ) , (4.6)where C = h T, T i and ǫ is a small positive constant to be determined later. Inserting(4.6) into (4.5), we have0 ≥ − r a − r − (24 + 4 ǫ ) · r ( a − r ) + ( 2 m − ǫ ) k H k − ǫC . (4.7)Now we shall use the technique in [19] to calculate the term ∆ r . In the case p = p ∗ , we denote a ∗ = r ( p , p ∗ ) > B a ∗ ( p ) k H k = k H k (˜ p ) . By Proposition 3.1 we have max B a ∗ ( p ) k H k = max ∂B a ∗ ( p ) k H k . a − r ( p ∗ )) k H k (˜ p ) = ( a − r (˜ p )) k H k (˜ p ) ≤ ( a − r ( p ∗ )) k H k ( p ∗ ) . (4.8)For any p ∈ B a ∗ ( p ), by the Gauss equation we know that the Ricci curvature Ric ( M, g )is bounded from below by R ii ( p ) ≥ − k H k (˜ p ) . (4.9)By the Laplacian comparison theorem, we get∆ r ≤ m + ( m − r k H k (˜ p ) . (4.10)Substituting (4.10) into (4.7), we have( 2 m − ǫ ) k H k ( p ∗ ) ≤ c ( m, ǫ ) a ( a − r ) + ǫC + 2( m − k H k (˜ p ) aa − r , (4.11)where c ( m, ǫ ) is a positive constant depending only on m and ǫ . Multiplying both sidesof (4.11) by ( a − r ) ( p ∗ ), we have( 2 m − ǫ )Ψ ≤ c ( m, ǫ ) a + ǫC a + 2( m − a ( a − r )( p ∗ ) k H k (˜ p ) , (4.12)where C = C , if C ≥ , if C < . By (4.8) and the inequality of Schwarz, we getΨ ≤ c ( m, ǫ ) a + mC ǫ − mǫ a . (4.13)In the case p = p ∗ , we have r ( p , p ∗ ) = 0. It is easy to get (4.13) from (4.7). Therefore(4.13) holds on B a ( p ).Then, at any interior point q ∈ B a ( p ), we get k H k ( q ) ≤ c ( m, ǫ ) a ( a − r ) + mC ǫ − mǫ a ( a − r ) . (4.14) Case 1. If C >
0, we choose ǫ sufficiently small, such that mC ǫ − mǫ ≤ λ . For a → ∞ , at the point q , we get k H k ≤ λ .
10n particular, k H k ( p ) ≤ λ . This contradicts to k H k = λ at p , so we can conclude that k H k ≡ M n .Therefore it is an affine m -plane by theorem 1. Case 2. If C = 0, let a → ∞ in (4.14), we get k H k = 0 . On the other hand, it is easy to see that H is a zero vector, since M is an m -dimensionalspacelike submanifold. Thus T ∈ T M . This contradicts to the translating vector T is nonzero. Then there exists no complete m -dimensional spacelike translating solitonwith a lightlike translating vector. Case 3. If C <
0, let a → ∞ in (4.14), we get k H k ≤ . This is impossible. Then there exists no complete m -dimensional spacelike translatingsoliton with a timelike translating vector. This completes the proof of Theorem 3. (cid:3) Remark : Here we mention that we can replace k H k with k B k in the proof oftheorem 3 and use proposition 3.2 to prove the second fundamental form k B k ≡
0. Itdirectly prove M m must be a plane. To gain theorem 5, we will show that the graph spacelike translating soliton is completewith respect to the induced metric g . Thus from theorem 3, we complete the proof oftheorem 5. Using the similar calculation of section 5 in [34], we prove the completenessof the induced metric g if k B k has an upper bound. To the simplicity, here we shalluse theorem 2.1 of Prof. Xin in [29] to obtain the completeness. Proposition 5.1:
Let u α be smooth functions defined everywhere in R m . Supposetheir graph M = ( x, u ( x )) is a spacelike translator in R m + nn . If the norm of the secondfundamental form k B k has a bound, then M is complete with respect to the inducedmetric. Proof:
Without loss of generality, we assume that the origin 0 ∈ M . FromProposition 3.1 of [20], we know that the pseudo-distance function z = h X, X i on M is a non-negative proper function. By (2.6), we know that if k B k has an upper bound,11 H k also has an upper bound. On the other hand, by (3.1) we get k∇ ⊥ H k = −h∇ ⊥ H, ∇ ⊥ H i = − X j h X k h T, e k i B jk , X l h T, e l i B jl i≤ | V | k B k ≤ ( | T | + k H k ) k B k . Then k∇ ⊥ H k also has a bound. By Theorem 2.1 in [29], we have for some k >
0, theset { z ≤ k } is compact, then there is a constant c depending only on the dimension m and the bounds of mean curvature and its covariant derivatives such that for all x ∈ M with z ( x ) ≤ k , |∇ z | ≤ c ( z + 1) . (5.1)Let γ : [0 , r ] → M be a geodesic on M issuing from the origin 0. Integrating (5.1) gives z ( γ ( r )) + 1 ≤ exp( cr ) , which forces M to be complete with respect to the induced metric. (cid:3) In the following we prove the bound of k B k . Proposition 5.2:
Under the assumption in Theorem 5, the function k B k has anupper bound on R m . Proof:
By assumption in theorem 5, there exists a large enough R such that( g ij ) > ǫ | x | I in case | x | > R . Choose a positive number a > R . Let B a (0) := { x ∈ R m | | x | ≤ a } . Consider the function below F ( x ) := ( a − | x | ) k B k defined on B a (0). Obviously, F attains its supremum at some interior point p ∗ . Wecan assume that k B k ( p ∗ ) >
0. Then, at p ∗ ,(log F ) i = 0 , (log F ) ij ≤ . In the following we calculate the first equation explicitly, and contract the second termabove with the positive definite matrix ( g ij ).( k B k ) i k B k − x i a − | x | = 0 , (5.2) n X i,j =1 g ij (cid:18) ( k B k ) ij k B k − ( k B k ) i ( k B k ) j k B k − x i x j ( a − | x | ) − δ ij a − | x | (cid:19) ≤ . (5.3)12rom Proposition 3.2 and the inequality of Schwarz, we get X i,j g ij ( k B k ) ij k B k = ∆ k B k k B k + X g ij Γ kij ( k B k ) k k B k ≥ n k B k − h V, ∇k B k k B k i + X i,j,k g ij Γ kij ( k B k ) k k B k ≥ n k B k − mn | V | − mn ( ∇k B k ) k B k + X i,j,k g ij Γ kij ( k B k ) k k B k . (5.4)In the following we shall estimate the term P i,j,k g ij Γ kij ( k B k ) k k B k . By (2.10) (2.11)and (5.2), we have X i,j,k g ij Γ kij ( k B k ) k k B k = X α ( a i u αi − b α ) g kl u αl x k a − | x | . By the inequality of Cauchy, we get X α ( a i u αi − b α ) g kl u αl x k ≤ "X α ( a i u αi − b α ) "X α ( g kl u αl x k ) . By the spacelike of M , we have X α ( u αi ) < , ∀ ≤ i ≤ m. Then "X α ( a i u αi − b α ) ≤ "X α ( b α ) + "X α ( X i a i u αi ) ≤| ~b | + " | ~a | ( X α X i ( u αi ) ) = | ~b | + | ~a | "X i ( X α u αi ) ≤| ~b | + √ m | ~a | , where ~a = ( a , ..., a m ) , ~b = ( b , ..., b n ). On the other hand,13 X α ( g kl u αl x k ) ≤ "X α ( g kl u αl u αk ) ( g kl x l x k ) ≤ " ( X i g ii )( X α ( X k ( u αk ) )) " ( X k g kk ) | x | ≤ " ( X i g ii )( X k ( X α ( u αk ) )) ( X k g kk ) | x |≤√ ma ( X i g ii ) . Therefore X i,j,k g ij Γ kij ( k B k ) k k B k ≤ C a P g ii a − | x | , (5.5)where C = 4 m ( | ~a | + | ~b | ).Thus, inserting (2.6), (5.2) and (5.5) into (5.4), we get X i,j g ij ( k B k ) ij k B k ≥ n k B k − mn | V | − mn ( ∇k B k ) k B k − C a P g ii a − | x | ≥ n k B k − mn C − mn a P g ii ( a − | x | ) − C a P g ii a − | x | , (5.6)where C = h T, T i .Then, inserting (5.2) and (5.6) into (5.3), we have k B k ≤ m C + (16 mn + 28 n ) a P g ii ( a − | x | ) + nC a P g ii a − | x | . (5.7)Multiplying both sides of (5.7) by ( a − | x | ) ( p ∗ ), we have( a − | x | ) k B k ( p ∗ ) ≤ m C a + [(16 mn + 28 n ) a + nC a ] X g ii . (5.8)If p ∗ ∈ B R (0), then for any x ∈ B a (0), F ( x ) ≤ a max B R (0) k B k . If p ∗ ∈ B a (0) \ B R (0), by assumption X g ii ≤ m | x | ǫ , we get [( a − | x | ) k B k ]( p ∗ ) ≤ m C a + C a + mnC ǫ a . (5.9)14here C is a positive constant depending only on m , n and ǫ . Thus, in both cases, wealways have F ( x ) ≤ [( a − | x | ) k B k ]( p ∗ ) ≤ C a + C a , (5.10)where C is a positive constant depending only on m , n , ǫ , C , C and max B R (0) k B k .Then, at any interior point q ∈ B a (0), we obtain[( a − | x | ) k B k ]( q ) ≤ [( a − | x | ) k B k ]( p ∗ ) ≤ C a + C a . (5.11)Dividing both sides of (5.11) by ( a − | x | ) , we obtain k B k ( q ) ≤ C a ( a − | x | ) + C a ( a − | x | ) . (5.12)Let a → ∞ in (5.12), we get k B k ( q ) ≤ C . This completes the proof of Proposition 5.2. (cid:3)
Acknowledgements : We wish to express our sincere gratitude to Prof. Qun Chenand Xingxiao Li for thier valuable and helpful discussions on the topic.